Final Fantasy IV used a very unique approach when it came to creating a party of characters for the player to control: party members moved in and out of the party based solely on the demands of the story, rather than through direct player control. While this approach takes some control over the game away from the player, I think it can allow for a game with a much more personal story than is seen in many RPGs.
There are two approaches to RPG party construction that tend to be much more common than FF IV's approach. In the first, as seen in games like Dragon Quest VIII or Wild ARMS 3, the party consists of a handful of characters who join early and are never swapped out. In the second major method, as seen in every Final Fantasy from VI on, the player has a large pool of characters, of which only a fraction can be in the party at a time, but can be freely swapped in and out. The second approach in particular is almost a standard feature of RPGs. However, these approaches force the game developers to write the plot of a game in a particular way, limiting the kind of plots that are possible.
The second approach's main problem can be very visible: the game developers can rarely customize the game's story and events for specific party compositions. No matter which characters are involved in an event, the event will have to resolve the same way. Let's look at Chrono Trigger as an example. Chrono Trigger gave each character in the team custom dialogue for every scene. However, the characters in the game were only playing out generic roles in each scene. For example, after the Ocean Palace, one of the characters in the party takes over a leadership position and asks the local elder some questions. No matter which party member is placed in that role, the same general information is communicated to the player, particularly since the NPC dialogue is inflexible. So even in Chrono Trigger, one of the best games at handling a swappable party in history, can at best offer only minor variations in dialogue and minor additions to scenes based on party composition. In games with larger casts, such as Final Fantasy VI, developers often have to resort to using generic dialogue. In many cases, developers do not even give dialogue to characters who can be swapped out. The result of this overall approach is that it mandates that the story has to be more about the team as a whole than the individual party members.
The set-up where there is a single unchanging party can avoid the problem of generic dialogue and uncustomized scenes, but does require certain plot considerations. Most notably, the entire party typically has to be introduced within the first several areas of the game. Furthermore, once the party is together, it can no longer be separated for any long period of time. A good example is Wild ARMs 4: the first three party members all join in the opening segment of the game, while the fourth joins soon afterwards. Once the four characters are together, the party is only broken up for a couple short scenarios before it reunifies. This actually produces similar results to the previous approach: the stories of the individual members who make up the group are subsumed into the story of the team as a whole, which can potentially limit the growth of individual characters. The game need to put the party together could also result in rather forced introductions in games (in other words: "Why are all of the characters from the same place when the world is so huge?").
The strength of the approach where party members join and leave freely based on plot is that it makes very few demands on writers. The only demand that it does make is that party members join and leave the party under plausible conditions. However, the act of a party member joining or leaving can create a lot of drama in a game. Tellah's confrontation with Edward in Damcyan castle, Leviathan's attack, Palom and Porom's sacrifice, Rydia's dramatic return in the underworld, and so on are all plot points created to justify the appearance or disappearance of a character, but they include most of Final Fantasy IV's most memorable and dramatic scenes. This also puts the focus of the game's plot on the individual stories of the characters. Tellah is a good example: he joins Cecil at first when their paths go the same way, but leaves after his daughter's death to pursue revenge on his own. When Tellah returns to the party, he joins only to get a chance at taking his revenge, which he ends up sacrificing his life for. The entire time Tellah is a part of the story, his tale of loss and revenge is an ever-present undercurrent in the game that is never completely subsumed by Cecil's journey of redemption. The plot-based approach opens up the opportunity for specific characters to introduce their own sub-plots, develop them, and then bow out of the main story when their own plots are resolved, without ever having to pace out their stories based on the flow of the game as a whole.
The biggest weakness of Final Fantasy IV's story-based approach to party membership is that it is not well suited to game mechanics that involve long-term character customization. It would be really disappointing for a player if he sinks hours into carefully building up a character only to watch that character leave after the next plot event. However, the story approach does work well with game systems like FF VII's Materia system or FF VIII's Junction system, which allow the player to customize characters by equipping them with abilities kept in a party-wide pool. I am actually rather surprised that I have never seen a game that combined such a system with plot-based party membership.
I would like to see more games that used Final Fantasy IV's way of doing things. It is a system that has never seen as much use over the years as it deserves.